EULER'S

Ian Bruce

This is another project
involving one of Euler's favorite topics : the finding of curves satisfying
maximum or minimum properties under various circumstances ; in the first
chapter, a general method of finding
such curves is set out and numerous examples of its use presented. In the second
and third chapters, the work examines increasingly complicated applications of
what was later called the Calculus of Variations. At first determined functions
are treated, in which the values of the derivatives of the curve are known at
any point; later more abstract constructions are introduced in chapters 2 &
3 where Euler deals with functionals related by a general integral formula; for
example, the theory developed can be used to find the shape of the curve a body
can fall along in the minimum time subject to various forms of resistance, etc.
Chapter 4, section 7 sets out the precepts for the 5 kinds of extrema
considered ; this is a very useful summary.

It
is of course a fundamental work in the establishing the Calculus of Variations
by Euler and Lagrange a little later, being a sort of hybrid of ordinary
integration and the solving of differential equations satisfying certain
conditions ; the functions being integrated are usually not algebraic or
transcending, but involve differential formulas that may be expanded out in
terms of the usual *x, y , p, q, r *,
etc. , and so special methods are needed to solve such problems. A good place
to look initially for a modern appraisal of the work is the article by Craig G.
Fraser, '*The Origins of Euler's Variational
Calculus' in the Archives of the Exact Sciences*. Vol. 47, No. 2 (June
1994), pp. 103-141. Pub. by Springer.
However, Chapter 2 of Herman H. Goldstine's classic work ' *A History of the Calculus of Variations,
from the 17 ^{th} through the 19^{th} Century*', pub. by
Springer in 1980, must take pride of place, and should be consulted regularly
with this translation, although of course in one chapter only a glimpse of the
contents of Euler's book can be set out. Other places to look include of course
Wikipedia ; the book by Routh on

Click here for the 1^{st} Chapter :*
**Concerning the Method
of finding the maxima and minima of curved lines generally**.*

This is an introductory
chapter in which most of what lies ahead in the following chapters is surveyed
perhaps in a rather abstract view. Thus, Euler distinguishes between the 'ordinary' max. and min. of a given curve,
and the present situation, where the curve itself is to be varied in a given
domain to produce an extreme value. A great deal of careful argument is gone
into, the relevance of which may only become apparent on reading further
chapters. The use of an integral *Zdx*
as the formula carrying the formula was revolutionary at the time, as was the
use of *x*, *y, p, q*, etc. as variables; a most convenient notation, not involving
derivatives directly greater than first order. It is a good idea to
return to this introductory chapter and to refresh oneself on the ideas
presented. See H. Goldstine p.67 onwards for a review of this chapter and the
next.

Click here for
Chapter 2*a* :* **A Method
for finding the absolute maxima and minima for curved lines.*

This is a long chapter containing much material, which I have split
into two parts for computing convenience. Here Euler has been able to set out
the conditions for the max. or min. of a curve contained within an integral in
a straight forwards manner, and depending on *y, p, q*, etc.. The basic
idea consists of extending some arbitrary *y* coordinate of the max. or
min. curve by the infinitesimal amount *nv*, calculating the changes
arising in *dZ* at neighbouring points and equating these changes to zero,
so allowing Euler to set up a necessary condition for a max. or min. in terms
of simple differential equations, depending on the form of dZ , and into an
increasing number of terms. Numerous examples
are given, the work is continued in part 2*b*. At present determinate
formulas only are considered.

Click here for
Chapter 2*b* :* **A Method
for finding the absolute maxima and minima for curved lines.*

Click here for
Chapter 3 :** On finding max./min. …. with indeterminate magnitudes present. ** This is a long chapter in which the
mathematical procedures needed to resolve various physical problems are set out
; such problems involve bodies sliding down unknown curves according to some
law of resistance, with the aim of minimizing the time of falling, etc. Thus a
number of unknown functions are present in the integrand to be minimized. This
leads Euler to an extension of his original scheme, so that the calculations
become more involved, but still follow the same general lines. See H. Goldstine
p.73 onwards for a review of this chapter.

Click here for
Chapter 4 :** Concerning the use of the method now treated in the resolution of
various kinds of questions. ** This
is another long chapter ; however, in section 7 Euler has summarized the 5 main
methods he has set up in chapter 3 for finding optimal curves under various
conditions ; the examples help make the work of chapter 3 much easier to
understand; one wonders why he took so long to introduce his examples….. See H.
Goldstine p.84 onwards for a review of this chapter.

Click here for
Chapter 5 : *A method of finding that curve
among all the curves with a given property, which may be endowed with the
property of being a max. or min.*

** ** This is another long chapter ; here Euler
considers the introduction of an extra condition to be satisfied by the max. or
min. curve, and satisfied by all the curves :

Click here for
Chapter 6 : *A method for determining that
curve among the curves endowed with several common properties, which may be provided with the
max. or min. property.*

** ** In this final chapter, before moving on to
appendices, Euler examines the problem of variation when some number of common
conditions are present, and the max. min. value of a curve is to be found
satisfying these conditions. The third proposition lands Euler in trouble when
he tries to generalize and to produce the same result extended by his new line
of reasoning, that he found before from the general method introduced.
Nevertheless, the old method works, and is used to solve three more examples.

Click here for
Appendix 1A : *The curves associated with
elastic laminas.*

** ** In this appendix, later added to the main work,
Euler sets out to show that his method of finding maxima or minima curves
associated with generalized functions in the form of integrals, can be applied to finding the shape of loaded
laminas or ribbons, as had then recently been established in a straightforward
method from mechanics by Daniel Bernoulli, following on the earlier work of his
uncle, James Bernoulli. Most of the first part of this appendix, so subdivided
for convenience, is given over to finding the nine classes or kinds of shapes
adopted by a flexed lamina under different end conditions. An English
translation exists already in Isis (1933) by Oldfather et al., of which I have
just become aware, and have not referred to here.

Click here for
Appendix 1B : *The curves associated with
elastic laminas cont'd.*

** ** This completes the deflections and vibrations
of beams as presented by Euler at this time; he was to correct an error spotted
by Daniel Bernoulli, which we will deal with here soon.

Click here for
Appendix II : *The motion of projectiles in a
non-resisting medium, determined by the method of maxima and minima .*

** ** This completes Euler's work at this time; in
this last section is the germ of the calculus of variations; Euler is able to
derive the equations of motion of projectiles and orbits according to the

Click here for E296 : *The elements of the calculus of variations.*

** ** Soon after Lagrange sent Euler a letter and
his method for shortening Euler's calculations in this treatise, Euler produced
this paper in which he sets out Lagrange's contributions in a clear manner.

Click here for E297 : *The analytical explanation of the method of
maxima and minima.*

** ** In this second paper, Euler revisits his
formulas and rewrites his main results in term of variations; this is probably
the best place to look if you do not have the desire or the time to read the
above book. The most satisfactory thing of course is to do both.

Ian
Bruce. Aug. 8th ,
2013 latest revision. Copyright : I reserve the right to publish this
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